11.19
11.20
ψ 1 ( y) = N 1 (2y)e
−y 2 /2
11.21 (a)
(b) V(x) is continuous at L/2.
a
L
b
L
cL
a b c
2
2
2
4
8
4 8
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
+ =
or
−
+
=
≥
Z
2
2
2
2
2
2
m x
x
cL
x
E x
x
L
d
d
/
ψ
ψ
ψ
( )
( )
( )
−
+
+
=
≤
Z
2
2
2
2
3
2m x
x
a x
bx
x
E x
x
d
d
ψ
ψ
ψ
( )
(
) ( )
( )
L/2
E
m
m
mk
k
m
1
2
2
2
2
1 2
3
2
3
2
3
2
/
=
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
Z
Z
Z
Z
α
⎛ ⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
1 2
3
2
/
Zω
−
+ =−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ + =
Z
Z
Z
2
4
2
2
2
2
2
2
0
m
k
m
mk
k
α
ψ
α
α
1
2
2
4
2
2
1
2
3
2
( )
x x
m
k
m
E
−
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
−
⎛
⎝
Z
Z
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
= 0
−
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ +
Z
2
1
2
4
2
2
2
2
3
m
x
x
k x
ψ
α
α
( )
2 2
2
1
1 1
x
x
E x
ψ
ψ
( )
( )
=
=
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
−
( )
/
2
2
3
2
1
2
3
4
2
1
2
2
N e
x
x
x
x
α
α
α
ψ
α
x x
2
4
2
3
α
α
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
=
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ + −
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
− /
2
2
1
2
2
2
2
2
2
N e
x
x
x
α
α
α
α
− −
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
+ −
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
−
2
2
2
2
2
2
2
2
x e
x e
x
x
α
α
α
/
/ α α 2
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
d
d
d
d
2
2 1
1
2
2
2
2
2
2
x
x
N
x
e
x
x
ψ
α
α
α
( )
/
=
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
−
/
e
x
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
2
2
2α
d
d
d
d
x
x
N
x
xe
N xe
x
x
ψ
α
α
α
α
1
1
2
1
2
2
2
2
2
2
2
( )
/
/
=
=
−
−
− −
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
−
x
e
x
α
α
2
2
2
2
/
−
+
=
Z
2
2
2
2
2
2
m x
x
k x x
E x
d
d
ψ
ψ
ψ
( )
( )
( )
2
0
2
2
2
2
(
)
/
N e
x
x
A
−
∞
∫
α
d
ANSWERS TO PROBLEMS
465
9781405124362_5_end.qxd 4/29/08 9:17 Page 465